It has been shown that the set of universal functions on trees contains a linear subspace except zero, dense in the space of harmonic functions. In this paper we show that the set of universal ...functions contains two linear subspaces except zero, dense in the space of harmonic functions that intersect only at zero. We work in the most general case that has been studied so far, letting our functions take values over a topological vector space.
Universal series in ∩ p >1 ℓ p Koumandos, S.; Nestoridis, V.; Smyrlis, Y.-S. ...
The Bulletin of the London Mathematical Society,
02/2010, Volume:
42, Issue:
1
Journal Article
Universal series in ∩p>1ℓp Koumandos, S.; Nestoridis, V.; Smyrlis, Y.‐S. ...
The Bulletin of the London Mathematical Society,
02/2010, Volume:
42, Issue:
1
Journal Article
Peer reviewed
In this paper an abstract condition is given yielding universal series defined by sequences a = {aj}∞j=1 in ∩p>1ℓp but not in ℓ1. We obtain a unification of some known results related to ...approximation by translates of specific functions including the Riemann ζ-function, or a fundamental solution of a given elliptic operator in ℝν with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in ℝν simultaneously with respect to all σ-finite Borel measures in ℝν. Stronger results are obtained by using universal Dirichlet series.
We consider the spaces \(H_{F}^{\infty}(\Omega)\) and \(\mathcal{A}_{F}(\Omega)\) containing all holomorphic functions \(f\) on an open set \(\Omega \subseteq \mathbb{C}\), such that all derivatives ...\(f^{(l)}\), \(l\in F \subseteq \mathbb{N}_0=\{ 0,1,...\}\), are bounded on \(\Omega\), or continuously extendable on \(\overline{\Omega}\), respectively. We endow these spaces with their natural topologies and they become Fréchet spaces. We prove that the set \(S\) of non-extendable functions in each of these spaces is either void, or dense and \(G_\delta\). We give examples where \(S=\varnothing\) or not. Furthermore, we examine cases where \(F\) can be replaced by \(\widetilde{F}=\{ l\in \mathbb{N}_0:\min F \leqslant l \leqslant \sup F\}\), or \(\widetilde{F}_0= \{ l\in \mathbb{N}_0:0\leqslant l \leqslant \sup F\}\) and the corresponding spaces stay unchanged.
Recently, harmonic functions and frequently universal harmonic functions on a tree \(T\) have been studied, taking values on a separable Fr\'{e}chet space \(E\) over the field \(\mathbb{C}\) or ...\(\mathbb{R}\). In the present paper, we allow the functions to take values in a vector space \(E\) over a rather general field \(\mathbb{F}\). The metric of the separable topological vector space \(E\) is translation invariant and instead of harmonic functions we can also study more general functions defined by linear combinations with coefficients in \(\mathbb{F}\). Unlike the past literature, we don't assume that \(E\) is complete and therefore we present a new argument, avoiding Baire's theorem.
If a Jordan curve {\sigma} has a one-sided conformal collar with "good" properties, then, using the Reflection principle, we show that any other conformal collar of {\sigma} from the same side has ...the same "good" properties. A particular use of this fact concerns analytic Jordan curves, but in general the Jordan arcs we consider do not have to be analytic. We show that if an one-sided conformal collar bounded by {\sigma} is of class A^p, then any other collar bounded by {\sigma} and from the same side of {\sigma} is of class A^p.
Smooth Universal Taylor Series Kariofillis, Ch; Konstadilaki, Ch; Nestoridis, V.
Monatshefte für Mathematik,
3/2006, Volume:
147, Issue:
3
Journal Article
We prove that universal approximation (uniform approximation on compact subsets with connected complement) implies almost everywhere approximation in the sense of Menchoff with respect to any given ...σ-finite Borel measure on
(d≥2).
Using a recent Mergelyan type theorem for products of planar compact sets we establish generic existence of Universal Taylor Series on products of planar simply connected domains Omegai, i=1, . . . , ...d. The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets Ki, i=1, . . . , d such that the complement of Ki is connected and for at least one i0 the set Ki0 is disjoint from Omegai0.
We show that the set of frequently universal harmonic functions on a tree T contains a vector space except 0 which is dense in the space of harmonic functions on T seen as subset of C^T . In order to ...prove this we replace the complex plane C by any separable Frechet space E and we repeat all the theory.
